Introduction to Environmental Modeling

Introduction

Thus far in developing the conservation equations for mass, chemical species, momentum, and mechanical energy, we have considered forms that apply either at the megascale in terms of integrals over the domain (e.g. Eq. (10.10), (11.43), (12.26), and (12.82)) where no gradients of properties within the system are considered or at the microscale (e.g. Eq. (10.34), (11.50), (12.31), and (12.75)) where point-to-point spatial variability within the system is modeled. In fact, these two alternatives can be considered extremes in modeling scenarios. Between these extremes are cases where spatial variability is of interest in one or two coordinate directions while the quantities are averaged over the other directions. We will refer to these kinds of models as being mixed scale because they involve both megascopic and microscopic scales.

To obtain mixed-scale models, we start with the general form of the equation in integral form. The formulation of any model then requires selection of a control volume that is consistent with the variability we wish to account for. A volume encompassing the entire system provides a fully megascale model. A volume that, in the limit, approaches a point provides a microscopic model. We can pick a volume that spans a study region in two spatial dimensions and is microscopic in the third dimension. For example, if we are modeling flow in a river, we might identify a control volume that covers the cross section of the river, but is infinitesimal in the direction normal to the river channel. Such a volume leads to equations that model properties that have been averaged over the river cross section as they vary along the river channel. Another mixed-scale model is one that integrates over one spatial domain but allows for variation of the averages over that domain in the two remaining coordinate directions. For example, mass and momentum conservation equations known as the shallow-water flow equations are obtained by integrating over the depth of flow in shallow regions while modeling the average velocity field and the depth of flow as a function of the lateral coordinates.